US6538585B2ExpiredUtilityA1
Distance-enhancing coding method
Est. expiryOct 4, 2020(expired)· nominal 20-yr term from priority
Inventors:Pi-Hai Liu
H03M 13/31H03M 5/145
51
PatentIndex Score
6
Cited by
6
References
7
Claims
Abstract
The present invention pertains to a distance-enhancing coding method that can be applied to digital recording and digital communications. It improves the time-varying maximum transition run method used in a conventional distance-enhancing coding to avoid main error events ±(1,-1) from happening. Under the premise of maintaining a code gain of 1.8 dB, the code rate can be increased from ¾ to ⅘. The invention also provides a method of using an enumeration algorithm and an exhaustive method to search for block codes for distance-enhancing coding, which can find required codes by following specific steps.
Claims
exact text as granted — not AI-modifiedWhat is claimed is:
1. A distance-enhancing coding method for receiving input sequences and converting an m-bit received input sequence into a corresponding n-bit codeword following specific conversion rules, with m and n being positive integers and n>m, so that the codewords satisfy constraints characterized in that:
the constraints comprise a number of primary constraints and b number of exceptional constraints with a and b being positive integers, the primary constraint being a constraint on the number of consecutive code symbols with a same value and the exceptional constraint being a constraint on a segment of code sequences starting at a specific position satisfying an exceptional form, so that codewords satisfying either the primary constraints or the exeptional constraints are considered as satisfying the constraints;
wherein the primary constraint and the exceptional constraint comprise:
the number of consecutive 1s is not greater than k 1 ;
the number of consecutive 1s starting at an odd position is not greater than k 1,odd ;
the number of consecutive 1s starting at an even position is not greater than k 1,even ;
the number of consecutive 0s is not greater than k 0 ;
the number of 0s before the first 1 is not greater than λ; and
the number of 0s after the last 1 is not greater than ρ; wherein k 1 , k 1,odd , k 1,even , k 0 , λ and ρ are integers;
further wherein the exceptional form of the exceptional constraint is to add to ‘10’ consecutively k, 1s follows by ‘01’ and the exceptional form starts at an even position if k 1,odd >k 1,even while the exceptional form starts at an odd position if k 1,odd <k 1,even .
2. The method of claim 1 , wherein k 1 =2, k 1,odd =2, k 1,even =1 in the primary constraints and the exceptional constraint is that ‘101101’ starts at an even position.
3. The method of claim 2 , wherein m=8 and n=10.
4. The method of claim 3 , wherein k 0 =11, ρ=6, and λ=5.
5. The method of claim 2 , wherein k 0 =11 in the primary constraints.
6. The method of claim 5 , wherein ρ=6 and λ=5 in the primary constraints.
7. The method of claim 2 , wherein the codewords are the 256 codewords:
0000010000
0001101000
0100010001
0101101001
1000010010
1001101010
0000010001
0001101001
0100010010
0101101010
1000010100
1001101101
0000010010
0001101010
0100010100
0101101101
1000010101
1010000001
0000010100
0001101101
0100010101
0110000001
1000010110
1010000010
0000010101
0010000001
0100010110
0110000010
1000011000
1010000100
0000010110
0010000010
0100011000
0110000100
1000011001
1010000101
0000011000
0010000100
0100011001
0110000101
1000011010
1010000110
0000011001
0010000101
0100011010
0110000110
1000100000
1010001000
0000011010
0010000110
0100100000
0110001000
1000100001
1010001001
0000100000
0010001000
0100100001
0110001001
1000100010
1010001010
0000100001
0010001001
0100100010
0110001010
1000100100
1010010000
0000100010
0010001010
0100100100
0110010000
1000100101
1010010001
0000100100
0010010000
0100100101
0110010001
1000100110
1010010010
0000100101
0010010001
0100100110
0110010010
1000101000
1010010100
0000100110
0010010010
0100101000
0110010100
1000101001
1010010101
0000101000
0010010100
0100101001
0110010101
1000101010
1010010110
0000101001
0010010101
0100101010
0110010110
1000101101
1010011000
0000101010
0010010110
0100101101
0110011000
1001000000
1010011001
0000101101
0010011000
0101000000
0110011001
1001000001
1010011010
0001000000
0010011001
0101000001
0110011010
1001000010
1010100000
0001000001
0010011010
0101000010
0110100000
1001000100
1010100001
0001000010
0010100000
0101000100
0110100001
1001000101
1010100010
0001000100
0010100001
0101000101
0110100010
1001000110
1010100100
0001000101
0010100010
0101000110
0110100100
1001001000
1010100101
0001000110
0010100100
0101001000
0110100101
1001001001
1010100110
0001001000
0010100101
0101001001
0110100110
1001001010
1010101000
0001001001
0010100110
0101001010
0110101000
1001010000
1010101001
0001001010
0010101000
0101010000
0110101001
1001010001
1010101010
0001010000
0010101001
0101010001
0110101010
1001010010
1010101101
0001010001
0010101010
0101010010
0110101101
1001010100
1010110100
0001010010
0010101101
0101010100
0110110100
1001010101
1010110101
0001010100
0010110100
0101010101
0110110101
1001010110
1010110110
0001010101
0010110101
0101010110
0110110110
1001011000
1011010000
0001010110
0010110110
0101011000
1000000001
1001011001
1011010001
0001011000
0100000001
0101011001
1000000010
1001011010
1011010010
0001011001
0100000010
0101011010
1000000100
1001100000
1011010100
0001011010
0100000100
0101100000
1000000101
1001100001
1011010101
0001100000
0100000101
0101100001
1000000110
1001100010
1011010110
0001100001
0100000110
0101100010
1000001000
1001100100
1011011000
0001100010
0100001000
0101100100
1000001001
1001100101
1011011001
0001100100
0100001001
0101100101
1000001010
1001100110
1011011010
0001100101
0100001010
0101100110
1000010000
1001101000
0001100110
0100010000
0101101000
1000010001
1001101001.
8 .The method of claim 2 wherein the codewords are transposes of the 256 codewords:
0000010000
0001101000
0100010001
0101101001
1000010010
1001101010
0000010001
0001101001
0100010010
0101101010
1000010100
1001101101
0000010010
0001101010
0100010100
0101101101
1000010101
1010000001
0000010100
0001101101
0100010101
0110000001
1000010110
1010000010
0000010101
0010000001
0100010110
0110000010
1000011000
1010000100
0000010110
0010000010
0100011000
0110000100
1000011001
1010000101
0000011000
0010000100
0100011001
0110000101
1000011010
1010000110
0000011001
0010000101
0100011010
0110000110
1000100000
1010001000
0000011010
0010000110
0100100000
0110001000
1000100001
1010001001
0000100000
0010001000
0100100001
0110001001
1000100010
1010001010
0000100001
0010001001
0100100010
0110001010
1000100100
1010010000
0000100010
0010001010
0100100100
0110010000
1000100101
1010010001
0000100100
0010010000
0100100101
0110010001
1000100110
1010010010
0000100101
0010010001
0100100110
0110010010
1000101000
1010010100
0000100110
0010010010
0100101000
0110010100
1000101001
1010010101
0000101000
0010010100
0100101001
0110010101
1000101010
1010010110
0000101001
0010010101
0100101010
0110010110
1000101101
1010011000
0000101010
0010010110
0100101101
0110011000
1001000000
1010011001
0000101101
0010011000
0101000000
0110011001
1001000001
1010011010
0001000000
0010011001
0101000001
0110011010
1001000010
1010100000
0001000001
0010011010
0101000010
0110100000
1001000100
1010100001
0001000010
0010100000
0101000100
0110100001
1001000101
1010100010
0001000100
0010100001
0101000101
0110100010
1001000110
1010100100
0001000101
0010100010
0101000110
0110100100
1001001000
1010100101
0001000110
0010100100
0101001000
0110100101
1001001001
1010100110
0001001000
0010100101
0101001001
0110100110
1001001010
1010101000
0001001001
0010100110
0101001010
0110101000
1001010000
1010101001
0001001010
0010101000
0101010000
0110101001
1001010001
1010101010
0001010000
0010101001
0101010001
0110101010
1001010010
1010101101
0001010001
0010101010
0101010010
0110101101
1001010100
1010110100
0001010010
0010101101
0101010100
0110110100
1001010101
1010110101
0001010100
0010110100
0101010101
0110110101
1001010110
1010110110
0001010101
0010110101
0101010110
0110110110
1001011000
1011010000
0001010110
0010110110
0101011000
1000000001
1001011001
1011010001
0001011000
0100000001
0101011001
1000000010
1001011010
1011010010
0001011001
0100000010
0101011010
1000000100
1001100000
1011010100
0001011010
0100000100
0101100000
1000000101
1001100001
1011010101
0001100000
0100000101
0101100001
1000000110
1001100010
1011010110
0001100001
0100000110
0101100010
1000001000
1001100100
1011011000
0001100010
0100001000
0101100100
1000001001
1001100101
1011011001
0001100100
0100001001
0101100101
1000001010
1001100110
1011011010
0001100101
0100001010
0101100110
1000010000
1001101000
0001100110
0100010000
0101101000
1000010001
1001101001.
9. The method of claim 1 , wherein k 1 =2, k 1,odd =1, k 1,even =2 in the primary constraints and the exceptional constraint is that ‘101101’ start at an odd position.
10. The method of claim 9 , wherein m=8 and n=10.
11. The method of claim 9 , wherein k 0 =11 in the primary constraints.
12. The method of claim 11 , wherein ρ=6, and λ=5 in the primary constraints.
13. The method of claim 1 , wherein k 1 =2, k 1,odd =1, k 1,even =2 in the primary constraints and the exceptional constraint is that ‘1011101’ starts at an odd position.
14. The method of claim 1 , wherein k 1 =3, k 1,odd =3, k 1,even =2 in the primary constraints and the exceptional constraint is that ‘1011101’ starts at an even position.
15. The method of claim 1 , wherein k 1 =3, k 1,odd =3, k 1,even =2 in the primary constraints and the exceptional constraint is that ‘1011101’ starts at an odd position.
16. The method of claim 1 , wherein the method for searching codewords comprises the steps of:
(step 1) specifying a block length of n and a code rate m/n for the code;
(step 2) setting ρ to be an integer not greater than k 0 , enumerating the binary sequences with a length of i and satisfying the ( k 1,odd =2, k 1,even =1, k 0 , λ=∞, ρ) constraints, and denoting the number by t 1 ;
(step 3) setting λ= k 0 −ρ so that there will be N n −t n-λ-1 code patterns satisfying the ( k 1,odd =2, k 1,even =1, k 0 , λ= k 0 −ρ, ρ) constraints. The sequences obtained by concatenating these N n −t n-λ-1 code patterns arbitrarily will satisfy the ( k 1 odd =2, k 1,even =1, k 0 ) constraints;
(step 4) enumerating the sequences which satisfy the exceptional constraints. The number will be B n = ∑ j = 1 ⌊ n 6 ⌋ ( X n , j - Y n , j ) ,
where L is the length of the exceptional constraint, X n , j = ∑ ( i 1 , i 2 , i 3 , i j ) ≥ 0 , 2 ∑ l = 1 j i l ≤ n - jL [ ( N n - jL - 2 ∑ l = 1 j t l + 1 ) ∏ l = 1 J ( t 2 i l + 1 ) ] , and Y n , j = ∑ ( i 1 , i 2 , i 3 , i j ) ≥ 0 , 2 ∑ l = 1 j i l ≤ n - jL - λ - 1 [ ( N n - λ - 1 - jL - 2 ∑ l = 1 j t l + 1 ) ∏ l = 1 J ( t 2 i l + 1 ) ] ;
(step 5) C n = B n + N n −t n-λ-1 as the total number of binary sequences of length n and satisfying the ( k 1,odd , k 1,even , k 0 )+(exceptional constraint) constraints, where the exceptional constraint is described in claim 3 . The sequences obtained by concatenating these C n code patterns arbitrarily will satisfy the ( k 1,odd =2, k 1,even =1, k 0 )+(exceptional constraint) constraints;
(step 6) adjusting n, k 0 , and m and returning to step 1 if C n is less than 2 m , and selecting 2 m codewords from the C n codewords to form a code with a code rate of m/n if C n is not less than 2 m .
17. The method of claim 16 , wherein the step 2 in the method for searching codewords performs enumeration according to the following definitions:
(D.0) two binary sequences of length n are said to be
X =( xn, . . . , x 2, x 1) Y =( yn, . . . , y 2, y 1)( xp>yp ) and ( xi=yi )∀ p<i≦n:
The sequence Y is said to be ordered before the sequence X.
(D.1) An is the set of (k 1,odd , k 1,even , k 0 , λ=∞, ρ) constrained sequences of length n;
(D.2) R(X) is the number of sequences Y εAn such that XY;
(D.3) R(0)=0 where 0 is a binary sequence with n 0s;
(D.4) res(X) is the sequence obtained by modifying the first nonzero bit of X to be zero;
(D.5) U i is the minimum sequence among sequences in An and has the first nonzero symbol at position i;
(D.6) M i is the maximum sequence among sequences in An and has the first nonzero symbol at position i; R ( X _ ) = ∑ i = 0 n - 1 x i w i ; (D.7)
(D.8) t i =R(M i ); and
(D.9) N i = w i +t i-2 .
18. The method of claim 16 , wherein the step 2 in the method for searching codewords sets ρ to be the largest integer that is not less than k 0 /2.
19. The method of claim 16 , wherein the step 6 in the method for searching codewords comprises the step of increasing n, k 0 and then returning the step 1 if C n <2 m .
20. The method of claim 16 , wherein the step 6 in the method for searching codewords comprises the step of decreasing m and then returning the step 1 if C n <2 m .
21. The method of claim 16 , wherein the step 6 in the method for searching codewords comprises the step of forming a code with a code rate of m/n by selecting 2 m codewords from the C n codewords, the selecting criteria is to minimize the parameter k 0 if C n is not less than 2 m .
22. The method of claim 16 , wherein the step 6 in the method for searching codewords comprises the step of forming a code with a code rate of m/n by selecting 2 m codewords from the C n codewords, the selecting criteria is to prevent the quasi-catastrophic error events to occur if C n is not less than 2 m .
23. A distance-enhancing coding method for receiving input sequences and converting the input sequences according to specific conversion rules into codewords to prevent dominated error events from occurring, the codewords being obtained through the following steps:
(step 1) Let A be the set of sequences with the number of consecutive code symbol 1s less than or equal to k 1 and the number of consecutive code code symbol 0s less than or equal to k 0 , where k 1 is the minimum integer that will make sequences in A have dominated error events;
(step 2) B is a set which consist of sequences in set A with the maximum number of consecutive 1s exactly k 1 ;
A sequence in set B is classified into the set Boo if the numbers of consecutive 0s before and after a segment of consecutive k 1 1s are both odd;
A sequence in set B is classified into the set Bee if the numbers of consecutive 0s before and after a segment of consecutive k 1 1s are both even;
A sequence in set B is classified into the set Boe if the numbers of consecutive 0s before and after a segment of consecutive k 1 1s are odd and even, respectively;
A sequence in set B is classified into the set Beo if the numbers of consecutive 0s before and after a segment of consecutive k 1 1s are even and odd, respectively;
Boo, Bee, Boe, and Beo are referred to as the oo-constrained set, ee-constrained set, oe-constrained set, and eo-constrained set, respectively.
(step 3) Select one or a number of plural of the sets Boo, Bee, Boe, and Beo. Set C is the union of the selected sets. Subtract the set C from the set A to form a set D, i.e., D consists of sequences which is in A but not in C simultaneously. The set D contains the desired codewords.
24. The method of claim 23 , wherein k 1 =2 when the dominated error event set is Σ={e 1 =(1,−1), e 2 =(−1,1)}.
25. The method of claim 24 , wherein the step 3 selects the sets Boo and Boe to form set C.
26. The method of claim 24 , wherein the step 3 selects the set Boe to form set C.
27. The method of claim 24 , wherein the step 3 selects the set Beo to form set C.
28. The method of claim 23 , wherein k 1 =3 when the dominated error event set is Σ={e 1 =(1,−1,1), e 2 =(−1,1,−1)}.
29. The method of claim 28 , wherein the step 3 selects the sets Bee and Beo to form set C.
30. The method of claim 28 , wherein the step 3 selects the sets Boo and Boe to form set C.
31. The method of claim 28 , wherein the step 3 selects the set Boe to form set C.
32. The method of claim 28 , wherein the step 3 selects the set Beo to form set C.
33. A distance-enhancing coding method, which can avoid the occurrence of s types of dominated error events in the error event set Σ={e 1 , e 2 , . . . , e s } and require the number of continuous 0s not greater than k 0 in the coded result for receiving input sequences and converting an m-bit received input sequences into a corresponding n-bit codeword following specific conversion rules, with m and n being positive integers, the method for searching the codewords comprises the following steps:
(step 1) Let A n be the set of sequences of length n and with the number of consecutive code symbol 1s less than k 1 and the number of consecutive code symbol 0s less than or equal to k 0 , where k 1 is the maximum integer that will make the sequences in A n free from dominated error events;
(step 2) Let B n be the set of binary sequences of length n and with the number of the longest consecutive 1s in the sequences being k 1 , the number of consecutive 0s being less than or equal to k 0 , and satisfying the θ , τ, λ and ρ constraints, where θ is the maximum number of 1s before the first 0, τ is the maximum number of 1s after the last 0, λ is the maximum number of 0s before the first 1, and ρ is the maximum number of 0s after the last 1. Taking into account of the fact that one needs to exclude dominated error events even at the border of codeword connections, we choose θ = └k 1 /2┘ , i.e., the maximum integer not greater than k 1 /2,τ= k 1 − θ ;
(step 3) Select a subset C n of B n using the exhaustive method so that no dominated error events in Σ can occur within any C n . The number of codewords contained in C n is preferably as big as possible. The following sub-procedure gives the way to achieve this goal:
(3.0) Initially, let C n and E n be equal to the empty set;
(3.1) Let x n be a code pattern in B n but not in the union of C n and E n ;
(3.2) Compute the error patterns of x n and each code pattern y n in C n . If no error pattern is a dominated error event □, then x n is included into C n ; otherwise, let E n include x n ;
(3.3) If the union of C n and E n is equal to B n , then stop this sub-procedure; otherwise, return to (3.1);
(step 4) Let D n be the union of A n and C n . If the number of code patterns in D n is not less than 2 m , then a code with a rate of m/n can be constructed by selecting 2 m code patterns from D n .
34. The method of claim 33 , wherein the dominated error event set Σ={e 1 =(1,−1), e 2 (−1,1)}, k 1 =2, θ=1, τ=1.
35. The method of claim 34 , wherein m=8 and n=10.
36. The method of claim 34 , wherein k 0 =9.
37. The method of claim 34 , wherein λ+ρ is not greater than k 0 .
38. The method of claim 37 , wherein λ=4 and ρ=5.
39. The method of claim 38 , wherein m=8, n=10, and k 0 =9.
40. The method of claim 37 , wherein λ=b 5 and ρ=4.
41. The method of claim 34 , wherein the codewords are any 256 codewords selected from the 257 codewords:
0000100000
0100000101
1000100010
0000110100
0100100110
1000101101
0000100001
0100001000
1000100100
0000110101
0100101101
1000110000
0000100010
0100001001
1000100101
0000110110
0100110001
1000110001
0000100100
0100001010
1000101000
0001000110
0100110100
1000110010
0000100101
0100010000
1000101001
0001001101
0100110101
1000110100
0000101000
0100010001
1000101010
0001010110
0100110110
1000110101
0000101001
0100010010
1001000001
0001011000
0101000110
1000110110
0000101010
0100010100
1001000010
0001011010
0101001101
1001000110
0001000001
0100010101
1001000100
0001101000
0101010110
1001001101
0001000010
0100100000
1001000101
0001101001
0101011000
1001010110
0001000100
0100100001
1001001000
0001101010
0101011010
1001011000
0001000101
0100100010
1001001001
0001101100
0101100000
1001011010
0001001000
0100100100
1001001010
0001101101
0101100010
1001101000
0001001001
0100100101
1001010000
0010000110
0101101000
1001101001
0001001010
0100101000
1001010001
0010001101
0101101001
1001101010
0001010000
0100101001
1001010010
0010010110
0101101010
1001101100
0001010001
0100101010
1001010100
0010011000
0101101100
1001101101
0001010010
0101000001
1001010101
0010011010
0101101101
1010000110
0001010100
0101000010
1010000001
0010100110
0110001000
1010001101
0001010101
0101000100
1010000010
0010101101
0110001001
1010010110
0010000001
0101000101
1010000100
0010110001
0110001010
1010011000
0010000010
0101001000
1010000101
0010110100
0110001100
1010011010
0010000100
0101001001
1010001000
0010110101
0110001101
1010100110
0010000101
0101001010
1010001001
0010110110
0110100000
1010101101
0010001000
0101010000
1010001010
0011000001
0110100001
1010110001
0010001001
0101010001
1010010000
0011000010
0110100010
1010110100
0010001010
0101010010
1010010001
0011000100
0110100100
1010110101
0010010000
0101010100
1010010010
0011000101
0110100101
1010110110
0010010001
0101010101
1010010100
0011000110
0110100110
1011000001
0010010010
1000000001
1010010101
0011010000
0110101000
1011000010
0010010100
1000000010
1010100000
0011010001
0110101001
1011000100
0010010101
1000000100
1010100001
0011010010
0110101010
1011000101
0010100000
1000000101
1010100010
0011010100
0110101101
1011000110
0010100001
1000001000
1010100100
0011010101
0110110000
1011010000
0010100010
1000001001
1010100101
0011010110
0110110001
1011010001
0010100100
1000001010
1010101000
0011011000
0110110010
1011010010
0010100101
1000010000
1010101001
0011011001
0110110100
1011010100
0010101000
1000010001
1010101010
0011011010
0110110101
1011010101
0010101001
1000010010
0000100110
0100000110
0110110110
1011010110
0010101010
1000010100
0000101101
0100001101
1000000110
1011011000
0100000001
1000010101
0000110000
0100010110
1000001101
1011011001
0100000010
1000100000
0000110001
0100011000
1000010110
1011011010
0100000100
1000100001
0000110010
0100011010
1000100110.
42. The method of claim 34 , wherein the codewords are the transposes of any 256 codewords selected from the 257 codewords:
0000100000
0100000101
1000100010
0000110100
0100100110
1000101101
0000100001
0100001000
1000100100
0000110101
0100101101
1000110000
0000100010
0100001001
1000100101
0000110110
0100110001
1000110001
0000100100
0100001010
1000101000
0001000110
0100110100
1000110010
0000100101
0100010000
1000101001
0001001101
0100110101
1000110100
0000101000
0100010001
1000101010
0001010110
0100110110
1000110101
0000101001
0100010010
1001000001
0001011000
0101000110
1000110110
0000101010
0100010100
1001000010
0001011010
0101001101
1001000110
0001000001
0100010101
1001000100
0001101000
0101010110
1001001101
0001000010
0100100000
1001000101
0001101001
0101011000
1001010110
0001000100
0100100001
1001001000
0001101010
0101011010
1001011000
0001000101
0100100010
1001001001
0001101100
0101100000
1001011010
0001001000
0100100100
1001001010
0001101101
0101100010
1001101000
0001001001
0100100101
1001010000
0010000110
0101101000
1001101001
0001001010
0100101000
1001010001
0010001101
0101101001
1001101010
0001010000
0100101001
1001010010
0010010110
0101101010
1001101100
0001010001
0100101010
1001010100
0010011000
0101101100
1001101101
0001010010
0101000001
1001010101
0010011010
0101101101
1010000110
0001010100
0101000010
1010000001
0010100110
0110001000
1010001101
0001010101
0101000100
1010000010
0010101101
0110001001
1010010110
0010000001
0101000101
1010000100
0010110001
0110001010
1010011000
0010000010
0101001000
1010000101
0010110100
0110001100
1010011010
0010000100
0101001001
1010001000
0010110101
0110001101
1010100110
0010000101
0101001010
1010001001
0010110110
0110100000
1010101101
0010001000
0101010000
1010001010
0011000001
0110100001
1010110001
0010001001
0101010001
1010010000
0011000010
0110100010
1010110100
0010001010
0101010010
1010010001
0011000100
0110100100
1010110101
0010010000
0101010100
1010010010
0011000101
0110100101
1010110110
0010010001
0101010101
1010010100
0011000110
0110100110
1011000001
0010010010
1000000001
1010010101
0011010000
0110101000
1011000010
0010010100
1000000010
1010100000
0011010001
0110101001
1011000100
0010010101
1000000100
1010100001
0011010010
0110101010
1011000101
0010100000
1000000101
1010100010
0011010100
0110101101
1011000110
0010100001
1000001000
1010100100
0011010101
0110110000
1011010000
0010100010
1000001001
1010100101
0011010110
0110110001
1011010001
0010100100
1000001010
1010101000
0011011000
0110110010
1011010010
0010100101
1000010000
1010101001
0011011001
0110110100
1011010100
0010101000
1000010001
1010101010
0011011010
0110110101
1011010101
0010101001
1000010010
0000100110
0100000110
0110110110
1011010110
0010101010
1000010100
0000101101
0100001101
1000000110
1011011000
0100000001
1000010101
0000110000
0100010110
1000001101
1011011001
0100000010
1000100000
0000110001
0100011000
1000010110
1011011010
0100000100
1000100001
0000110010
0100011010
1000100110.
43. The method of claim 34 , wherein the codewords are any 256 codewords selected from the 283 codewords:
0000000001
0010100001
1000001001
1010101001
0011011001
0110110110
0000000010
0010100010
1000001010
1010101010
0011011010
1000000110
0000000100
0010100100
1000010000
0000000110
0100000110
1000001101
0000000101
0010100101
1000010001
0000001101
0100001101
1000010110
0000001000
0010101000
1000010010
0000010110
0100010110
1000011000
0000001001
0010101001
1000010100
0000011000
0100011000
1000011010
0000001010
0010101010
1000010101
0000011010
0100011010
1000100110
0000010000
0100000000
1000100000
0000100110
0100100110
1000101101
0000010001
0100000001
1000100001
0000101101
0100101101
1000110001
0000010010
0100000010
1000100010
0000110001
0100110001
1000110100
0000010100
0100000100
1000100100
0000110100
0100110100
1000110101
0000010101
0100000101
1000100101
0000110101
0100110101
1000110110
0000100000
0100001000
1000101000
0000110110
0100110110
1001000110
0000100001
0100001001
1000101001
0001000110
0101000110
1001001101
0000100010
0100001010
1000101010
0001001101
0101001101
1001010110
0000100100
0100010000
1001000000
0001010110
0101010110
1001011000
0000100101
0100010001
1001000001
0001011000
0101011000
1001011010
0000101000
0100010010
1001000010
0001011010
0101011010
1001100000
0000101001
0100010100
1001000100
0001100000
0101100000
1001100010
0000101010
0100010101
1001000101
0001100010
0101100010
1001101000
0001000000
0100100000
1001001000
0001101000
0101101000
1001101001
0001000001
0100100001
1001001001
0001101001
0101101001
1001101010
0001000010
0100100010
1001001010
0001101010
0101101010
1001101100
0001000100
0100100100
1001010000
0001101100
0101101100
1001101101
0001000101
0100100101
1001010001
0001101101
0101101101
1010000110
0001001000
0100101000
1001010010
0010000110
0110000000
1010001101
0001001001
0100101001
1001010100
0010001101
0110000010
1010010110
0001001010
0100101010
1001010101
0010010110
0110001000
1010011000
0001010000
0101000000
1010000000
0010011000
0110001001
1010011010
0001010001
0101000001
1010000001
0010011010
0110001010
1010100110
0001010010
0101000010
1010000010
0010100110
0110001100
1010101101
0001010100
0101000100
1010000100
0010101101
0110001101
1010110001
0001010101
0101000101
1010000101
0010110001
0110100000
1010110100
0010000000
0101001000
1010001000
0010110100
0110100001
1010110101
0010000001
0101001001
1010001001
0010110101
0110100010
1010110110
0010000010
0101001010
1010001010
0010110110
0110100100
1011000001
0010000100
0101010000
1010010000
0011000001
0110100101
1011000100
0010000101
0101010001
1010010001
0011000100
0110100110
1011000101
0010001000
0101010010
1010010010
0011000101
0110101000
1011000110
0010001001
0101010100
1010010100
0011000110
0110101001
1011010000
0010001010
0101010101
1010010101
0011010000
0110101010
1011010001
0010010000
1000000000
1010100000
0011010001
0110101101
1011010010
0010010001
1000000001
1010100001
0011010010
0110110000
1011010100
0010010010
1000000010
1010100010
0011010100
0110110001
1011010101
0010010100
1000000100
1010100100
0011010101
0110110010
1011010110
0010010101
1000000101
1010100101
0011010110
0110110100
1011011000
0010100000
1000001000
1010101000
0011011000
0110110101
1011011001
1011011010.
44. The method of claim 34 , wherein the codewords are the transposes of any 256 codewords selected from the 283 codewords:
0000000001
0010100001
1000001001
1010101001
0011011001
0110110110
0000000010
0010100010
1000001010
1010101010
0011011010
1000000110
0000000100
0010100100
1000010000
0000000110
0100000110
1000001101
0000000101
0010100101
1000010001
0000001101
0100001101
1000010110
0000001000
0010101000
1000010010
0000010110
0100010110
1000011000
0000001001
0010101001
1000010100
0000011000
0100011000
1000011010
0000001010
0010101010
1000010101
0000011010
0100011010
1000100110
0000010000
0100000000
1000100000
0000100110
0100100110
1000101101
0000010001
0100000001
1000100001
0000101101
0100101101
1000110001
0000010010
0100000010
1000100010
0000110001
0100110001
1000110100
0000010100
0100000100
1000100100
0000110100
0100110100
1000110101
0000010101
0100000101
1000100101
0000110101
0100110101
1000110110
0000100000
0100001000
1000101000
0000110110
0100110110
1001000110
0000100001
0100001001
1000101001
0001000110
0101000110
1001001101
0000100010
0100001010
1000101010
0001001101
0101001101
1001010110
0000100100
0100010000
1001000000
0001010110
0101010110
1001011000
0000100101
0100010001
1001000001
0001011000
0101011000
1001011010
0000101000
0100010010
1001000010
0001011010
0101011010
1001100000
0000101001
0100010100
1001000100
0001100000
0101100000
1001100010
0000101010
0100010101
1001000101
0001100010
0101100010
1001101000
0001000000
0100100000
1001001000
0001101000
0101101000
1001101001
0001000001
0100100001
1001001001
0001101001
0101101001
1001101010
0001000010
0100100010
1001001010
0001101010
0101101010
1001101100
0001000100
0100100100
1001010000
0001101100
0101101100
1001101101
0001000101
0100100101
1001010001
0001101101
0101101101
1010000110
0001001000
0100101000
1001010010
0010000110
0110000000
1010001101
0001001001
0100101001
1001010100
0010001101
0110000010
1010010110
0001001010
0100101010
1001010101
0010010110
0110001000
1010011000
0001010000
0101000000
1010000000
0010011000
0110001001
1010011010
0001010001
0101000001
1010000001
0010011010
0110001010
1010100110
0001010010
0101000010
1010000010
0010100110
0110001100
1010101101
0001010100
0101000100
1010000100
0010101101
0110001101
1010110001
0001010101
0101000101
1010000101
0010110001
0110100000
1010110100
0010000000
0101001000
1010001000
0010110100
0110100001
1010110101
0010000001
0101001001
1010001001
0010110101
0110100010
1010110110
0010000010
0101001010
1010001010
0010110110
0110100100
1011000001
0010000100
0101010000
1010010000
0011000001
0110100101
1011000100
0010000101
0101010001
1010010001
0011000100
0110100110
1011000101
0010001000
0101010010
1010010010
0011000101
0110101000
1011000110
0010001001
0101010100
1010010100
0011000110
0110101001
1011010000
0010001010
0101010101
1010010101
0011010000
0110101010
1011010001
0010010000
1000000000
1010100000
0011010001
0110101101
1011010010
0010010001
1000000001
1010100001
0011010010
0110110000
1011010100
0010010010
1000000010
1010100010
0011010100
0110110001
1011010101
0010010100
1000000100
1010100100
0011010101
0110110010
1011010110
0010010101
1000000101
1010100101
0011010110
0110110100
1011011000
0010100000
1000001000
1010101000
0011011000
0110110101
1011011001
1011011010.
45. The method of claim 33 , wherein the dominated error event set Σ={e 1 =(1,−1,1), e 2 =(−1,1,1)}, k 1 =3, θ=1, τ=2.
46. The method of claim 45 , wherein 2 m codewords selected from the founded codewords are made into a codebook.
47. The method of claim 45 , wherein the transposes of 2 m codewords selected from the codewords are made into a codebook.Cited by (0)
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